Moment Of Inertia Calculator
Calculate moment of inertia calculator calculate rotational inertia I for solid cylinders, spheres, thin hoops, and rods from mass m and radius r. Accurate physics formulas and unit conversions for engineers, students & technicians.
Calculate Rotational Moment of Inertia (I)
Enter your physical parameters below to compute verified moment of inertia metrics.
Calculation Results
Calculated using verified physical methodology: Rotational Inertia: I = c \cdot m \cdot r^2
Radius of Gyration: k = \sqrt{\I / m}
Quick Summary
The Moment of Inertia Calculator computes rotational inertia ($I = c \cdot m \cdot r^2$) for solid cylinders, spheres, hoops, and rods across $\text{kg}\cdot\text{m}^2$, $\text{lb}\cdot\text{ft}^2$, and $\text{g}\cdot\text{cm}^2$.
Formula Explanation
Rotational Inertia: I = c \cdot m \cdot r^2
Radius of Gyration: k = \sqrt{\I / m}
How It Works
The Moment of Inertia Calculator applies geometric inertia coefficients ($c = 0.5$ for disk, $0.4$ for sphere, $1.0$ for hoop, $1/12$ for rod). It outputs rotational moment of inertia ($I$) and radius of gyration ($k = \sqrt{I/m}$).
Step-by-Step Worked Example
Practical Problem: Calculate the moment of inertia for a 50.0 kg industrial flywheel (solid disk) with radius $r = 0.4\text{ meters}$.
- Step 1: Identify Geometry & Input Values: Mass $m = 50.0\text{ kg}$, Radius $r = 0.4\text{ m}$, Shape = Solid Disk ($c = 0.5$).
- Step 2: Square Radius Parameter: $r^2 = (0.4)^2 = 0.16\text{ m}^2$.
- Step 3: Apply the Moment of Inertia Formula: $I = 0.5 \cdot m \cdot r^2$.
- Step 4: Execute Numeric Multiplication: $I = 0.5 \times 50.0\text{ kg} \times 0.16\text{ m}^2 = 4.000\text{ kg}\cdot\text{m}^2$.
- Step 5: Convert and Interpret Imperial Units & Radius of Gyration: Moment of Inertia $I = 4.000\text{ kg}\cdot\text{m}^2 = 94.94\text{ lb}\cdot\text{ft}^2 = 0.295\text{ slug}\cdot\text{ft}^2$. Radius of Gyration $k = \sqrt{4.0 / 50.0} = 0.2828\text{ meters}$.
Real-World Calculation Examples
Scenario 1: Industrial Flywheel Disk
Parameters: $m = 50\text{ kg}$, $r = 0.4\text{ m}$, Shape = Solid Disk
Result: $I = 4.000\text{ kg}\cdot\text{m}^2$ (94.94 lb·ft²). Engine flywheel inertia.
Scenario 2: Solid Bowling Ball
Parameters: $m = 6.0\text{ kg}$, $r = 0.11\text{ m}$, Shape = Solid Sphere
Result: $I = 0.0290\text{ kg}\cdot\text{m}^2$ (0.690 lb·ft²). Bowling ball sphere inertia.
Scenario 3: Bicycle Wheel Rim (Thin Hoop)
Parameters: $m = 1.5\text{ kg}$, $r = 0.35\text{ m}$, Shape = Thin Hoop
Result: $I = 0.1838\text{ kg}\cdot\text{m}^2$ (4.36 lb·ft²). Bicycle wheel hoop inertia.
Scenario 4: Thin Propeller Shaft Rod
Parameters: $m = 12.0\text{ kg}$, $L = 2.0\text{ m}$, Shape = Thin Rod
Result: $I = 4.000\text{ kg}\cdot\text{m}^2$ (94.94 lb·ft²). Propeller rod inertia.
Key Benefits of Using This Calculator
4 Standard Geometric Shapes
Supports solid disks, solid spheres, thin hoops, and thin rods about center.
Radius of Gyration ($k$)
Computes equivalent radius of gyration $k = \sqrt{I/m}$ for structural engineering.
Multi-Unit Readouts
Outputs inertia in $\text{kg}\cdot\text{m}^2$, $\text{lb}\cdot\text{ft}^2$, $\text{g}\cdot\text{cm}^2$, and $\text{slug}\cdot\text{ft}^2$.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is moment of inertia?
Moment of inertia (I) is the rotational analog of mass, measuring a rigid body's resistance to angular acceleration about a rotation axis (I = integral(r² dm)).
What is the formula for moment of inertia of a solid cylinder or disk?
I = 0.5 * m * r².
What is the formula for a solid sphere?
I = 0.4 * m * r² = (2/5) * m * r².
What is the formula for a thin hoop or ring?
I = m * r².
What is Parallel Axis Theorem?
The Parallel Axis Theorem I = I_cm + m * d² calculates moment of inertia about any axis parallel to a center-of-mass axis at distance d.
What is the SI unit of moment of inertia?
The SI unit is kilogram meter squared (kg·m²).
How converts kg·m² to lb·ft²?
Multiply kg·m² by 23.7304 to obtain lb·ft² (e.g. 4.0 kg·m² = 94.92 lb·ft²).
What is radius of gyration k?
Radius of gyration k = sqrt(I / m) is the radial distance at which the total mass could be concentrated to yield the same moment of inertia.
Why does mass distribution matter in rotation?
Mass located further from the axis of rotation contributes quadratically (r²) to moment of inertia, making objects harder to spin or stop spinning.
How relates moment of inertia to torque?
Newton's 2nd Law for rotation: tau = I * alpha, where alpha is angular acceleration.